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Thread: Two Dimensional Discrete Dynamical System

  1. #1
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    Two Dimensional Discrete Dynamical System

    Hi, I have the following dynamical system,

    $\displaystyle x_{n+1}={x_n}^2-{y_n}^2+a$
    $\displaystyle y_{n+1}=2x_ny_n$

    Where a is real.

    And I am asked to consider the set of points on a circle of radius r and centre origin and show that they are mapped to another circle under one iteration.

    How do I go about this? I know the equation for the circle is of course $\displaystyle x^2+y^2 = r^2$. Can someone please give me a clue how to start?

    Thanks,
    Katy
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  2. #2
    Junior Member erich22's Avatar
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    Re: Two Dimensional Discrete Dynamical System

    Let
    $\displaystyle x_0^2+y_0^2=r^2$ be the initial circle to be mapped by the system.
    Calculate $\displaystyle x_1,\text{ and } y_1$, then you can find that $\displaystyle x_1^2+y_1^2= \dots$, i.e. a circle with the origin is translated along x axis.

    There you go, Katy
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